I now understand! This is a very helpful answer. Thanks!

I'm having a trouble following the paragraph starting "This can only be no[n] constant if...". Would you mind adding a few details? What might be confusing me is that I might be misinterpreting the previous paragraph, so let me say a bit about how I am interpreting it. First, when you say "complex conjugation element of the Galois group" I assume that you mean "Galois group of $\overline{\mathbb{Q}}$", right? It then looks to me like you are fixing some such $\sigma$, and the overline in your big equation is the action of this $\sigma$.

@Venkataramana: That's a good point. Is there any simple description of what is going on here?

@WillSawin: It says that it is a lattice in an appropriate vector space, but you have to take some (nonalgebraic) logarithms before this becomes the case.